Method for evaluating electrical field levels in the near-zone of transmitting antennas

ABSTRACT

The method for evaluating electrical field levels in the near-zone of transmitting antennas, particularly linear array antenna widely used in cellular telephony systems, provides accurate evaluation while requiring reduced processing times and a restricted set of input data, such as physical external dimensions of the antenna, its gain and its radiation patterns in the vertical plane and in the horizontal plane.

Description

[0001] This invention relates to tools for planning radio carriertelecommunications systems and particularly relates to a method forevaluating electrical field levels in near-zones of transmittingantennas.

[0002] The increase of radio base stations (RBSs) for mobilecommunications in highly populated areas has caused growing anxiety inpeople living close to said stations, worried about biological effectsfrom electromagnetic pollution on the human body. As a consequence,governments in various countries have established stringent exposurelimits for the level of electromagnetic fields radiated in theradio-frequency range, in zones where the population may be exposed forlong periods of time. Mobile telephony operators must produce moredetailed documentation on the compliance of their radio base stations toprescribed limits. Consequently, the use of more accurate and reliableplanning tools is necessary to avoid costly and difficult interventionson devices and systems after construction and installation.

[0003] The electrical field radiated by transmitting antennas,particularly radio base station linear arrange antennas, is currentlycalculated under a far field hypothesis, expressing an antenna gain inspherical co-ordinates G(θ,φ) as the product of radiation patterns inthe horizontal and vertical planes.

[0004] The far field hypothesis is verified when the distance from theantenna exceeds $\frac{2\quad L_{0}^{2}}{\lambda},$

[0005] where L₀ is the maximum extension of the antenna and λ is thewavelength. Considering base radio station antennas and cellulartelephony system frequency bands, this distance can correspond to morethan 40 metres, while in urban areas base radio stations are usuallypositioned at less than 10 metres from inhabited buildings.Consequently, the use of more accurate methods for evaluating electricalfield levels in the near-zone of radio-frequency sources, particularlylinear array antennas, widely used in cellular telephony systems, isnecessary.

[0006] A large number of numerical methods and software tools for theevaluation of the electromagnetic field radiated by antennas arecurrently available. For example, the well-known software NEC-2,described in “Numerical Electromagnetic Code—Method of Moment”, by BurkG. J. and Poggio A. J., Lawrence Livermore National Laboratory, January1981, calculates the electromagnetic field very carefully, both in thefar-field and in the near-zone of the transmitting antenna, but requiresa good knowledge of wire-grid technique for input file implementation, aprecise mechanical quotation of all parts of the antenna and extensivecomputation time. In practice, however, the knowledge of the antenna islimited to its physical dimensions, its gain and its radiation patternsin the vertical and horizontal planes.

[0007] The method for evaluating electrical field levels in thenear-zone of transmitting antennas which is the object of the inventionovercomes said shortcomings and solves the described technical problemsby providing an accurate evaluation while requiring a limited number ofinput data and reduced processing time by data processing units used forcalculating.

[0008] Specifically, the object of the invention is a method forevaluating the electrical field levels in the near-zone of transmittingantennas, as described in the characterising part of claim 1.

[0009] Additional characteristics and advantages of the invention willnow be described, by way of example only, with reference to theaccompanying drawings wherein:

[0010]FIG. 1 schematically shows a linear array antenna arranged alongthe z axis of a Cartesian reference system in symmetric position withrespect to axis x;

[0011]FIG. 2 represents a thin wire of arbitrary length L arranged onthe z axis of a Cartesian reference systems in symmetric position withrespect to axis x.

[0012] As previously mentioned, a simple method for estimating theelectrical field radiated by a radio base linear array antenna is basedon the hypothesis that the antenna gain G(θ,φ) can be calculated as theproduct of radiation patterns in the vertical and horizontal planes.These patterns can be determined by means of measurements made in thefist phases of the procedure, along with the physical dimensionalmeasurements on the antenna. Alternatively, the patterns provided by themanufacturer can be used.

[0013] According to the method of the invention, the antenna gain G(θ,φ)is calculated in a slightly different way by introducing the angle Ψ,i.e.:

G(θ,φ)=G _(M) D _(V)(θ)D _(H)(Ψ)  (1)

[0014] where:

[0015] G_(M) is the maximum gain of the antenna

[0016] D_(V)(θ) is the radiation pattern in the vertical plane (φ=0)

[0017] D_(H)(Ψ) is the radiation pattern on a conical surface generatedby the rotation on axis z in the maximum radiation direction

[0018] angle Ψ, function of θ, φ and τ, results from:

Ψ(θ,φ,τ)=φ sin(β(θ,τ)), $\begin{matrix}{{{\psi \left( {\theta,\varphi,\tau} \right)} = {\varphi \quad \sin \quad \left( {\beta \quad \left( {\theta,\tau} \right)} \right)}},{{{where}\quad {\beta \left( {\theta,\tau} \right)}} = \left\{ \begin{matrix}{\theta \quad \frac{\pi}{\pi + {2\quad \tau}}} & {0 \leq \theta \leq {{\pi/2} + \tau}} \\{\left( {\pi - \theta} \right)\quad \frac{\pi}{\pi - {2\quad \tau}}} & {{{\pi/2} + \tau} < \theta \leq \pi}\end{matrix} \right.}} & (2)\end{matrix}$

[0019] where τ is the down-tilt of the radiation beam with respect tothe horizontal plane.

[0020] The angle Ψ is introduced to properly match the secondary lobesof the radiation pattern. Its analytical expression derives from settingthe following constraints:

[0021] The direction (θ,φ) does not depend on the angle φ for θ=0, π.This is because per θ=0 the direction (0,φ) is always the direction +z,while for θ=π the direction (0,φ) is always direction −z. Consequently,when θ=0,π the value of G(θ,φ) must not depend on the angle φ and,specifically, must be equal to the value of the vertical pattern onθ=0,π: because D_(H)(0)=1, the result is obtained by setting Ψ=0;

[0022] for θ=π/2+τ, the function G(θ,φ) must coincide with the radiationpattern in the horizontal plane. The result is obtained by setting Ψ=φbecause in θ=π/2+τ the function D_(v)(θ) has unitary value.

[0023] As known, the electrical field E(r,θ,φ) in the far-zone is givenby: $\begin{matrix}{{\underset{\_}{E}\left( {r,\theta,\varphi} \right)} = {\frac{\sqrt{30\quad P_{R}\quad {G\left( {\theta,\varphi} \right)}}}{r}{\hat{p}\left( {\theta,\varphi} \right)}}} & (3)\end{matrix}$

[0024] where P_(R) is the radiated power, r is the distance from theelectrical centre of the source and {circumflex over (p)} is thepolarisation vector.

[0025] We will now suppose that the separation of the effects in thevertical plane and in the horizontal plane, as expressed by equation(1), is still valid in the near-zone field, particularly in the zone 2to 3λ and $\frac{2\quad L_{0}^{2}}{\lambda},$

[0026] where L₀ is the maximum extension of the antenna. Because thephysical width of a linear array antenna is much smaller than itsheight, the horizontal pattern contribution D_(H)(Ψ) to the electricalfield in the near-zone field is approximately equal to that in afar-zone field.

[0027] According to this hypothesis, the near-zone electrical field canbe expressed as:

|E(r,θ,φ)|={square root}{square root over (P _(R) D_(H)(Ψ)}|E(r,θ)|  (4)

[0028] where F(r,θ) is an appropriate vectorial function which iscalculated as described below.

[0029] Equation 4 can consequently be used to calculate the electricalfield in a point P=(r,θ,φ) of the space as a product of twocontributions: the first of which is due to the electrical field in thehorizontal plane of the antenna D_(H)(Ψ) and the second of which is dueto the electrical field in the vertical plane of the antenna D_(V)(θ).

[0030] The second contribution is obtained by calculating the electricalfield in projection P₁=(r,θ,0) of point P=(r,θ,φ) on the vertical planex, z of the antenna, as shown in FIG. 1. In this figure, the lineararray antenna, schematically illustrated, is arranged along the axis zof a Cartesian reference system x, y, z in a symmetric position withrespect to axis x. Point P, identified by spherical co-ordinates r,θ,φ,is the point in space where the electrical field intensity is to beevaluated. Point P₁ is the projection of point P on the vertical planeof the antenna (plane x, z) and is obtained by turning point P, aroundaxis z, by an angle which is equal to the value of the sphericalco-ordinate 4.

[0031] To calculate the electrical field in P₁, we introduce the conceptof “Equivalent Current Distribution”, abbreviated as ECD, i.e. thedistribution of current which generates the same radiative effects of acertain source in the near-zone field and in the far-zone field.

[0032] In the case of a linear array antenna, which is verticallypolarised, the ECD can be conveniently defined as a one-dimensionalcurrent along a thin wire of arbitrary length L, as shown along axis zin symmetrical position with respect to axis x in the Cartesianreference system of FIG. 2.

[0033] In the figure, I(z) is the electrical current running in thewire, which is supposed polarised in direction z (i.e.I(z)=I(z){circumflex over (z)}). Vector r′ identifies a generic currentelement on the thin wire. Co-ordinates d, θ_(d), φ_(d)=0 are thespherical co-ordinates of point P₁ in a Cartesian reference system whoseorigin corresponds to the current element identified by vector r′.

[0034] ECD current is obtained from the knowledge of the radiationpattern in the vertical plane of the antenna: from the radiation theory,we can write the radiation pattern in terms of voltage V(θ) in thevertical plane x, z as: $\begin{matrix}{{V(\theta)} = {\sqrt{D_{V}(\theta)} = {C\quad \sin \quad \theta \quad {\int_{{- L}/2}^{{+ L}/2}{{I(z)}\quad {\exp \left( {j\quad 2\quad \pi \quad z\quad \cos \quad \theta} \right)}\quad {z}}}}}} & (5)\end{matrix}$

[0035] where C is a multiplying constant and I(z) is the currentgenerated in the wire.

[0036] By defining:

u=(cos θ)/λ,

{tilde over (V)}(u)=V(u)/{square root}{square root over (1−(λu)²)}

[0037] and resolving (5) with respect to I(z), i.e. to ECD, we obtain:$\begin{matrix}{{I(z)} = {{\int_{{- L}/2}^{{+ L}/2}{{\overset{\sim}{V}(u)}\quad \exp \quad \left( {{- j}\quad 2\quad \pi \quad z\quad u} \right)\quad {u}}} = {\quad \left\{ {\overset{\sim}{V}(u)} \right\}}}} & (6)\end{matrix}$

[0038] i.e., ECD is the Fourier transform of the previously definedradiation pattern {tilde over (V)}(u). It can be calculated with astandard Fourier Fast Transform (FFT), by using the Nyquist samplingtheorem and estimating the physical extension of the ECD as χ times thephysical height of the antenna. The resulting ECD is consequentlydiscrete, i.e. formed of a certain number of current samples I_(n).Numeric results demonstrate that χ values in the range from 2 to 3 areacceptable for linear arrays.

[0039] Alternatively, the ECD can be obtained with the Woodward-Lawsonsampling method (for example, see article by G. A. Somers “A proof ofthe Woodward-Lawson sampling method for a finite linear array”, RadioScience, Vol. 28, No. 4, pp. 481-485, July-August 1993).

[0040] The ECD determined in this way is a discrete currentdistribution, therefore the vectorial function F(r,θ), required tocompute the electrical field in the vertical plane according to (4), isgiven by: $\begin{matrix}{{\underset{\_}{F}\left( {r,\theta} \right)} = {j\quad \frac{Z_{0}}{2\quad \lambda}{\sum\limits_{n}{\frac{\exp \quad \left( {{- j}\quad n\quad k_{0}d} \right)}{d}I_{n}\quad \sin \quad \theta_{d}{\hat{\theta}}_{d}}}}} & (7)\end{matrix}$

[0041] where Z₀ is the characteristic impedance of vacuum, equal to 377Ω, λ is the wavelength, k₀=2π/λ is the constant propagation in vacuum,I_(n) is the nth ECD sample, and d and θ_(d), as previously mentioned,are the spherical co-ordinates of point P₁ in which to compute theelectrical field in the Cartesian reference centred in the nth currentsample I_(n), as shown in FIG. 2.

[0042] Naturally, numerous changes can be implemented to theconstruction and embodiments of the invention herein envisaged withoutdeparting from the scope of the present invention, as defined by thefollowing claims.

1. Method for evaluating electrical field levels |E(r,θ,φ)| in a pointP(r,θ,φ) in the near-zone of transmitting antennas, particularly lineararray antennas located in the origin of a Cartesian reference systemidentified by the orthogonal co-ordinates (x, y, z) and the sphericalco-ordinates (r,θ,φ), in which: the power (P_(R)) of the radio-frequencytransmitted by the antenna is established; the radiation pattern (D_(V))of the antenna in the vertical plane is determined (φ=0); the radiationpattern (D_(H)) of the antenna in the horizontal plane is determined;the physical height of the antenna is measured; characterised in thatsaid signal power (P_(R)) and said radiation patterns (D_(V), D_(H)) areused as inputs for data processing means to determine the electricalfield level according to the following expression: |E(r,θ,φ)|=,{squareroot}{square root over (P _(R) D _(H)(Ψ))}|E(r,θ)| where: D_(H)(Ψ) isthe radiation pattern on a conical surface generated by the rotationabout axis z in the maximum radiation direction; angle Ψ, function of θ,φ and τ, is given by:${{\psi \left( {\theta,\varphi,\tau} \right)} = {\varphi \quad \sin \quad \left( {\beta \quad \left( {\theta,\tau} \right)} \right)}},{{{where}\quad {\beta \left( {\theta,\tau} \right)}} = \left\{ \begin{matrix}{\theta \quad \frac{\pi}{\pi + {2\quad \tau}}} & {0 \leq \theta \leq {{\pi/2} + \tau}} \\{\left( {\pi - \theta} \right)\quad \frac{\pi}{\pi - {2\quad \tau}}} & {{{\pi/2} + \tau} < \theta \leq \pi}\end{matrix} \right.}$

τ being the down-tilt of the radiation beam with respect to thehorizontal plane (x, y); and F(r,θ) is a vectorial function resultingfrom:${\underset{\_}{F}\left( {r,\theta} \right)} = {j\quad \frac{Z_{0}}{2\quad \lambda}{\sum\limits_{n}{\frac{\exp \quad \left( {{- j}\quad n\quad k_{0}d} \right)}{d}I_{n}\quad \sin \quad \theta_{d}{\hat{\theta}}_{d}}}}$

where Z₀ is the characteristic impedance of vacuum, 1 is the wavelength,k₀ is the propagation constant in vacuum, In is the nth sample ofone-dimension current (ECD) along a thin wire of arbitrary length (L),which generates the same radiative effects of the antenna both in thenear-zone and the far-zone, and d and θ_(d) are the sphericalco-ordinates of the projection P₁=(r,θ,0) of said point P(r,θ,φ) in thevertical plane (x, z).
 2. Method according to claim 1, characterised inthat the following Fourier transform ℑ{{tilde over (V)}(u)} is used todetermine said one-dimensional current (ECD) by means of data processingmeans:${I(z)} = {{\int_{{- L}/2}^{{+ L}/2}{{\overset{\sim}{V}(u)}\quad \exp \quad \left( {{- j}\quad 2\quad \pi \quad z\quad u} \right)\quad {u}}} = {\quad \left\{ {\overset{\sim}{V}\quad (u)} \right\}}}$

cos(θ)/λ and {tilde over (V)}(u)/{square root}{square root over(1−(λu)²)}, V(u) being the voltage radiation pattern in the verticalplane (x, z), resulting from radiation theory of said radiation pattern(D_(V)).
 3. Method according to claim 2, characterised in that saidFourier transform is calculated by said data processing means with astandard Fourier Fast Transform (FFT), by using the Nyquist samplingtheorem and estimating the physical extension of said one-dimensionalcurrent (ECD) as χ times said physical height of the antenna, being χcompressed in the range from 2 to
 3. 4. Method according to claim 2,characterised in that said one-dimensional current (ECD) is calculatedby data processing means according to Woodward-Lawson sampling method.